A six-sided fair die is rolled. Obtain the mutual information between the random variable $X$ and the random variables $Y$ and $Z$, where:
- $X$: number of on the front face (side facing you)
- $Y$: number of on the top face
- $Z$: number of on the bottom face
To answer this question, should I be finding the mutual information between $X$ and ($Y$ given $Z$)?
$I(X; Y|Z) = H(X|Z) - H(X|Y,Z)$
Would the fact that the front face can be 1 of 6 options, and then top face would be 1 of 4 options adjacent to this front face, and then given this top face, we know with certainty what the bottom face is mean that $H(X|Y,Z) = 0$?
The question asks for $I(X;Y,Z)$ (if I understand it correctly).
From the chain rule it follows $$ I(X;Y,Z) = I(X;Y|Z) + I(X;Z). $$ Now, since knowledge of $Y$ implies knowledge of $Z$ and vice versa, it follows that $$ \begin{align} I(X;Y|Z)&=H(X|Z)-H(X|Y,Z)\\ &=H(X|Z)-H(X|Z)\\ &=0, \end{align} $$ that is, given the value of $Z$, no (further) information is gained about $X$ when $Y$ is observed. I leave it to you the computation of $I(X;Z)$.